Question

hi! pleas click the image, if possible, please explain how I can prove this trigonometric identity?

Answer 1

Given trigonometric identity to prove:

$\frac{\cos^2t+4\cos t+4}{\cos t+2}=\frac{2\sec t+1}{\sec t}$

Take Left hand side of the equation:

$\begin{gathered} \frac{\cos^2t+4\cos t+4}{\cos t+2}=\frac{\cos ^2t+2\times2\times\cos t+2^2}{\cos t+2} \\ =\frac{(\cos t+2)^2}{\cos t+2} \\ =\cos t+2 \end{gathered}$

Take Right hand side of the equation:

$\begin{gathered} \frac{2\sec t+1}{\sec t}=\frac{\frac{2}{\cos t}+1}{\frac{1}{\cos t}} \\ =\frac{\frac{2+\cos t}{\cos t}}{\frac{1}{\cos t}} \\ =\cos t+2 \end{gathered}$

So, we have LHS = RHS.

Hence, proved.